# Hausdorff Space

1. Jan 23, 2009

### Symmetryholic

1. The problem statement, all variables and given/known data

Show that X is Hausdorff if and only if the diagonal $$\Delta = \{x \times x | x \in X \}$$ is closed in $$X \times X$$.

2. Relevant equations

Definition of Hausdorff Space (T2) : A topological space in which distinct points have disjoint neighborhoods.

3. The attempt at a solution

2. Jan 24, 2009

### Unco

With so little work on your part shown to go by it's difficult to know where you're stuck.

If you haven't already, express the closedness of $$\Delta$$ in $$X\times X$$ (which I assume has the product topology) in terms of the openness of its complement.

Now have a go at proving each direction (neither is more difficult than the other) if you haven't already, and please show us your efforts.

3. Jan 24, 2009

### Symmetryholic

->
If X is Hausdorff, the diagonal $$\Delta$$ is closed in $$X\times X$$.

Assume X is Hausdorff. Now, we have two distinct points x, y and disjoint open sets U, V containing x, y, respectively. The basis element $$U \times V$$ containing $$(x,y) \in X \times X$$ should not intersect $$\Delta$$ by the assumption given to the Hausdorff property.
For every $$(x,y) \notin \Delta$$, we have a basis element in $$X \times X$$ containing (x,y), which does not intersect $$\Delta$$.
Thus, $$X \times X \setminus \Delta is open$$ and we conclude $$\Delta$$ is a closed set in $$X \times X$$ .

<-
If the diagonal $$\Delta$$ is closed in $$X\times X$$, X is Haudorff.

Supppose $$\Delta$$ is closed in $$X\times X$$. Then, $$X \times X \setminus \Delta$$ is open. Let $$(x,y) \in X \times X$$ and $$x \neq y$$. For $$(x,y) \notin \Delta$$, we have a basis element $$U \times V$$ in $$X \times X$$ containing (x, y).
We remain to show U and V are disjoint. Suppose on the contrary that U and V are not disjoint. Then, there is an element $$(z,z) \in X times X$$ which belongs to both U and V. Contradicting the fact that x and y are distinct.
Thus, X is Hausdorff.

4. Jan 24, 2009

### Unco

That's quite a leap from your previous post.

Now how 'bout you tackle your https://www.physicsforums.com/showthread.php?t=287038" without copying down the solution from an external source. It's the only way you'll learn topology.

Last edited by a moderator: Apr 24, 2017
5. Jan 24, 2009

### Symmetryholic

Can you please give me a link of an external source you mentioned?
It's my self study of topology (I am not even majoring in math) and I don't need to copy the external source to show you something to impress you. Rather, I just did for my self study purpose and asked an advice if someone finds an error in my attempt to the solution.

6. Jan 24, 2009

### Unco

Well, in that case, your work is quite error-free indeed! I would only suggest rephrasing picking an element $$(x,y)\not\in \Delta$$ as picking an element $$(x,y)\in (X\times X)\backslash \Delta$$. Now, on to your other problem!